Question

5 Chứng​ minh rằng​:

a) 5⁵-5⁴+5³​ chia hết​ cho 7 b) 7⁶+7⁵-7⁴​ chia hết​ cho 11

Answer 1

a, Ta có:

\(5^5-5^4+5^3=5^3.\left(5^2-5+1\right)=5^3.21\)

Vì \(5^3.21\) chia hết cho 7 nên \(5^5-5^4+5^3\) chia hết cho 7(đpcm)

b, Ta có:

\(7^6+7^5-7^4=7^4.\left(7^2+7-1\right)=7^4.55\)

Vì \(7^4.55\) chia hết cho 11 nên \(7^6-7^5+7^4\) chia hết cho 11(đpcm)

Chúc bạn học tốt!!!

Answer 2

a, \(5^5-5^4+5^3=5^3\left(5^2-5+1\right)=5^3.21⋮7\)

\(\Rightarrowđpcm\)

b, \(7^6+7^5-7^4=7^4\left(7^2+7-1\right)=7^4.55⋮11\)

\(\Rightarrowđpcm\)

Answer 3

a) \(5^5-5^4+5^3=5^3\left(5^2-5+1\right)=5^3.21⋮7\)

\(\Leftrightarrow5^5-5^4+5^3⋮7\)

b) \(7^6+7^5-7^4=7^4\left(7^2+7-1\right)=7^4.55⋮11\)

\(\Leftrightarrow7^6+7^5-7^4⋮11\)

Answer 4

Đặt:

\(S=5^5-5^4+5^3\)

\(S=5^3.5^2-5^3.5+5^3.1\)

\(S=5^3.25-5^3.5+5^3.1\)

\(S=5^3\left(25-5+1\right)=5^3.21=5^3.3.7⋮7\)

\(X=7^6+7^5-7^4\)

\(X=7^4.7^2+7^4.7-7^4.1\)

\(X=7^4.49+7^4.7-7^4.1\)

\(X=7^4\left(49+7-1\right)\)

\(X=7^4.55=7^4.5.11⋮11\)